Simplify and expand the following expression: $ \dfrac{5}{2y + 10}- \dfrac{1}{y - 6}- \dfrac{2}{y^2 - y - 30} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{5}{2y + 10} = \dfrac{5}{2(y + 5)}$ We can factor the quadratic in the third term: $ \dfrac{2}{y^2 - y - 30} = \dfrac{2}{(y + 5)(y - 6)}$ Now we have: $ \dfrac{5}{2(y + 5)}- \dfrac{1}{y - 6}- \dfrac{2}{(y + 5)(y - 6)} $ The least common multiple of the denominators is: $ 2(y + 5)(y - 6)$ In order to get the first term over $2(y + 5)(y - 6)$ , multiply by $\dfrac{y - 6}{y - 6}$ $ \dfrac{5}{2(y + 5)} \times \dfrac{y - 6}{y - 6} = \dfrac{5(y - 6)}{2(y + 5)(y - 6)} $ In order to get the second term over $2(y + 5)(y - 6)$ , multiply by $\dfrac{2(y + 5)}{2(y + 5)}$ $ \dfrac{1}{y - 6} \times \dfrac{2(y + 5)}{2(y + 5)} = \dfrac{2(y + 5)}{2(y + 5)(y - 6)} $ In order to get the third term over $2(y + 5)(y - 6)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{2}{(y + 5)(y - 6)} \times \dfrac{2}{2} = \dfrac{4}{2(y + 5)(y - 6)} $ Now we have: $ \dfrac{5(y - 6)}{2(y + 5)(y - 6)} - \dfrac{2(y + 5)}{2(y + 5)(y - 6)} - \dfrac{4}{2(y + 5)(y - 6)} $ $ = \dfrac{ 5(y - 6) - 2(y + 5) - 4} {2(y + 5)(y - 6)} $ Expand: $ = \dfrac{5y - 30 - 2y - 10 - 4}{2y^2 - 2y - 60} $ $ = \dfrac{3y - 44}{2y^2 - 2y - 60}$